The last equation is a differential equation of the second order which can be solved by the well-known methods of Differential Equations. We will, however, obtain the solution by a method which is more instructive and which may be called an experimental method.
It will be observed that 6 and its first two derivatives are added in equation (1); therefore 0 must be such a function of t that when it is differentiated with respect to the time the result is a function of the same type. The only known elementary functions which satisfy this condition are the circular and exponential functions. But since circular functions may be obtained from exponential functions* the solution of equation (1) may be expressed in the form
6 = aeP', (2)
where a and 0 are constants. Replacing 0 and its first two derivatives in equation (1) by their values, which are obtained from equation (2), we get
(/32 + 2 a/3 + b2) ae^ = 0.
Evidently one or both of the factors must vanish. When aeP* = 0, 0 = 0, which means that there is no motion. This is called a trivial solution. When the other factor vanishes we get _
Substituting these values of /3 in equation (2). we obtain the following particular solutions :
In order to obtain the general solution we multiply the particular solutions by constants and add them. Hence
* See Appendix Avn,